Unit Vectors and i, j Components

Write vectors with i and j, find magnitudes and unit vectors, build a vector of a given length or angle, and match components to find unknowns.

In the exam for Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 14 real past-paper questions below

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Meet i and j

A delivery drone is told: fly 3 km east, then 4 km north. Those two numbers are all it needs. In vectors, we give the two directions their own names: i and j.

Formula

i = (10), j = (01)

$i$ — one unit in the x-direction (often east).
$j$ — one unit in the y-direction (often north).

Any vector is some i's plus some j's. The numbers in front are its components.

Formula

a i + b j = (a b)

$a$ — the x-component: how far across.
$b$ — the y-component: how far up. A negative number means down.

Worked example

1
Write $2 i + 3 j + 2 (4 i - j)$ as one vector.
2
$2 (4 i - j) = 8 i - 2 j$ — Multiply each component by 2.
3
$(2 + 8) i + (3 - 2) j$ — Add the i parts together and the j parts together.
4
$= 10 i + j$ — As a column vector this is 10 over 1.

From the position vectors lesson: if A and B have position vectors $a$ and $b$ , then $A B = b - a$ .

Rule

Use correct notation. By hand, underline a vector letter (a) or put an arrow over two points, as in $A B$ . Write a column vector with no line between the numbers — it is not a fraction.

Key idea

i and j are just column vectors in disguise. Add, subtract and multiply them part by part.

Magnitude

The magnitude of a vector is its length. The components are the two shorter sides of a right-angled triangle, so Pythagoras gives the length.

Formula

∣ a i + b j ∣ = a^{2} + b^{2}

$a$ — the x-component.
$b$ — the y-component. Squaring makes a negative component positive.

Rule

Learn this formula. 0606, 4037 and 4PM1 do not give it on the paper.

Worked example

1
A is $(1, 2)$ and B is $(7, 10)$ . Find $∣ A B ∣$ .
2
$A B = (7 - 1) i + (10 - 2) j = 6 i + 8 j$ — End minus start: b − a.
3
$∣ A B ∣ = 6^{2} + 8^{2} = 100$ — Pythagoras on the components.
4
$= 10$ — The length of AB.

A common exam question puts a letter in a component and gives the length. Squaring both sides turns it into a quadratic.

Worked example

1
$P Q = (p - 1) i + 4 j$ and $∣ P Q ∣ = 5$ . Find the possible values of p.
2
$(p - 1)^{2} + 4^{2} = 5^{2}$ — Magnitude squared equals the length squared.
3
$(p - 1)^{2} = 9$ — Subtract 16.
4
$p - 1 = \pm 3$ — Square root: there are two signs.
5
$p = 4 or p = - 2$ — Both values give a vector of length 5.

Common mistake

Writing $∣3 i - 4 j ∣ = 9 - 16$ . Square each component: $(- 4)^{2} = 16$ , so it is $9 + 16 = 5$ .

Rule

No calculator on Paper 1 for 0606 and 4037: leave surds simplified, for example $∣2 i - 4 j ∣ = 20 = 25$ .

Key idea

Magnitude means length. Square both components, add, square root.

Unit vectors

A unit vector has length 1. It keeps a direction but forgets the size, like an arrow on a compass.

To make one, divide a vector by its own length. The direction stays the same and the length becomes exactly 1.

Formula

\hat{a} = \frac{a}{∣ a ∣}

$a$ — the vector that gives the direction.
$∣ a ∣$ — its magnitude. Dividing by it shrinks or stretches $a$ to length 1.
$\hat{a}$ — read “a hat”: the unit vector in the direction of a.

Worked example

1
Find the unit vector in the direction of $3 i - 4 j$ .
2
$∣3 i - 4 j ∣ = 9 + 16 = 5$ — First find the length.
3
$\frac{1}{5} (3 i - 4 j)$ — Divide the whole vector by 5.
4
$= \frac{3}{5} i - \frac{4}{5} j$ — Both components are divided.
5
$(\frac{3}{5})^{2} + (\frac{4}{5})^{2} = \frac{25}{25} = 1$ — Check: its length is 1.

Common mistake

Dividing only one component by the length. The whole vector is divided, so both components change.

Key idea

Unit vector = vector ÷ its own length. Same direction, length 1.

A given length and direction

Questions often ask for a vector with a given length, pointing in a given direction. Find how many times longer it must be, then scale.

Formula

v = k \times \frac{d}{∣ d ∣}

$k$ — the length you want.
$d$ — any vector in the right direction. $\frac{d}{∣ d ∣}$ is its unit vector.

Worked example

1
Find the vector of magnitude 26 in the direction of $5 i - 12 j$ .
2
$∣5 i - 12 j ∣ = 25 + 144 = 13$ — The length of the direction vector.
3
$\frac{26}{13} = 2$ — We need a vector twice as long.
4
$2 (5 i - 12 j) = 10 i - 24 j$ — Scale both components by 2.

Worked example

1
$O R$ has magnitude $310$ and the same direction as $i + 3 j$ . Find $O R$ .
2
$∣ i + 3 j ∣ = 1 + 9 = 10$ — The length of the direction vector.
3
$\frac{3 10}{10} = 3$ — The surds cancel. No calculator needed.
4
$O R = 3 i + 9 j$ — Scale both components by 3.

Common mistake

Multiplying by the length you want without dividing first. $26 (5 i - 12 j)$ has length $26 \times 13$ , not 26.

Key idea

Length you want ÷ length you have = the number to multiply by.

Components from an angle

Sometimes you are given a length and an angle instead of components. Splitting the vector into i and j parts is called resolving it.

Formula

v = r cos θ i + r sin θ j

$r$ — the length (magnitude) of the vector.
$θ$ — the angle measured from the positive x-direction, anticlockwise.

Worked example

1
A vector has magnitude 10 and makes an angle of $3 0^{\circ}$ with the positive x-axis. Write it in i, j form.
2
$10 cos 3 0^{\circ} = 10 \times \frac{3}{2} = 53$ — The i part uses cos, the side next to the angle.
3
$10 sin 3 0^{\circ} = 10 \times \frac{1}{2} = 5$ — The j part uses sin, the side opposite the angle.
4
$v = 53 i + 5 j$ — Exact values, so it works without a calculator.

If the question gives $tan α$ instead of the angle, the tan tells you a direction vector directly: $tan α = \frac{up}{across}$ .

Worked example

1
A vector of magnitude 15 makes an angle $α$ above the positive x-axis, where $tan α = \frac{3}{4}$ . Find it.
2
$4 i + 3 j$ — A direction vector: 4 across and 3 up gives $tan α = \frac{3}{4}$ .
3
$∣4 i + 3 j ∣ = 5$ — Its length.
4
$\frac{15}{5} = 3 \Rightarrow 12 i + 9 j$ — Scale by 3 to get length 15.

Common mistake

Swapping sin and cos. With the angle measured from the x-axis, cos goes with i and sin goes with j.

Key idea

Length and angle in, components out: $r cos θ$ across, $r sin θ$ up.

Equating like vectors

Two vectors are equal only if they match in both directions. So equal vectors have equal i parts and equal j parts. This gives two equations.

Rule

If $p i + q j = r i + s j$ , then $p = r$ and $q = s$ . This is called equating like vectors.

Worked example

1
Find $λ$ and $μ$ such that $λ (2 i + j) + μ (- i + 3 j) = 7 i + 14 j$ .
2
$i : 2 λ - μ = 7$ — Collect the i parts on the left and match them with 7.
3
$j : λ + 3 μ = 14$ — Collect the j parts and match them with 14.
4
$6 λ - 3 μ = 21 \Rightarrow 7 λ = 35$ — Multiply the first equation by 3 and add the second.
5
$λ = 5, μ = 3$ — Then μ = 2(5) − 7 = 3.
6
$5 (2 i + j) + 3 (- i + 3 j) = 7 i + 14 j$ — Check: 10 − 3 = 7 and 5 + 9 = 14.

4PM1 questions often use two non-parallel vectors a and b instead of i and j. The same rule works: match the a parts and match the b parts.

In longer geometry questions you write one point two ways and then equate like vectors. That is the vector geometry lesson.

Common mistake

Matching only one component. One equation cannot find two unknowns. Write both the i equation and the j equation.

Key idea

One vector equation in two dimensions gives two ordinary equations.

Which tool? Read the question

Read the question to see which tool it wants.

You see	You do
“Find the magnitude” of a vector	Write it as $p i + q j$ , then use Pythagoras (below)
A letter in a component and a given length	Square both sides, solve the quadratic, keep both values unless told otherwise
“Unit vector in the direction of …”	Divide by its own magnitude
“Magnitude k in the direction of …”	Unit vector times k
A length and an angle (or $tan α$ )	Resolve: $r cos θ i$ $+ r sin θ j$
Unknowns on both sides of a vector equation	Equate the i parts and the j parts

For $∣ A B ∣$ : $A B = b - a$ , then $∣ p i + q j ∣$ $= p^{2} + q^{2}$ .

Common mistake

Giving a unit vector that is not length 1. Always check: square the components, add, and you should get 1.

Key idea

Every vector question here is length, direction or matching parts. Name which one you need before you start.

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